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Inertia of non-integer Hadamard powers of a non-negative matrix. (English) Zbl 1497.15006

Summary: We prove that, if \(A=[a_{ij}]\) is a conditionally negative definite matrix with all entries positive and \(f:(0,\infty)\to (0,\infty)\) is a non-constant operator monotone function, then \([f(a_{ij})]\) is also conditionally negative definite. Moreover, \([f(a_{ij})]\) is invertible if \(A\) is invertible. Next, we prove that, if \(A=[a_{ij}]\) is a symmetric matrix with all entries positive and only one positive eigenvalue, and \(f(t)=t^r\) for all \(t\in (0,\infty)\), \(r\in (0,1]\), then \([f(a_{ij})]\) also has only one positive eigenvalue and it is invertible if \(A\) is invertible. Analogous results are also considered when the diagonal entries of \(A\) are all zero and off-diagonal entries are all positive, which extends a result of R. Reams [Linear Algebra Appl. 288, No. 1–3, 35–43 (1999; Zbl 0933.15006)].

MSC:

15A09 Theory of matrix inversion and generalized inverses
15A18 Eigenvalues, singular values, and eigenvectors
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory

Citations:

Zbl 0933.15006
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References:

[1] Bapat, R., Multinomial probabilities, permanents and a conjecture of Karlin and Rinott, Proc Amer Math Soc, 102, 3, 467-472 (1988) · Zbl 0647.60019 · doi:10.1090/S0002-9939-1988-0928962-9
[2] Reams, R., Hadamard inverses, square roots and products of almost semidefinite matrices, Linear Algebra Appl, 288, 35-43 (1999) · Zbl 0933.15006 · doi:10.1016/S0024-3795(98)10162-3
[3] Bhatia, R., Positive Definite Matrices (2007), Princeton (NJ): Princeton University Press, Princeton (NJ) · Zbl 1133.15017
[4] Horn, R.; Johnson, CR., Matrix Analysis (2013), Cambridge: Cambridge University Press, Cambridge · Zbl 1267.15001
[5] Bapat, R.; Raghavan, TES., Nonnegative Matrices and Applications (1997), Cambridge: Cambridge University Press, Cambridge · Zbl 0879.15015
[6] Bhatia, R., Matrix Analysis (1997), New York: Springer-Verlag, New York
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